Non-Uniform Lattices of Large Systole Containing a Fixed 3-Manifold Group

Let d be a square free positive integer and Q(√d) a totally real quadratic field over Q. We show there exists an arithmetic lattice L in SL(8,R) with entries in the ring of integers of Q(√d) and a sequence of lattices Γₙ commensurable to L such that the systole of the locally symmetric finite volume manifold Γₙ SL(8,R) SO(8) goes to infinity as n → ∞, yet every Γₙ contains the same hyperbolic 3-manifold group Π, a finite index subgroup of the arithmetic hyperbolic 3-manifold vol3. Notably, such an example does not exist in rank one, so this is a feature unique to higher rank lattices.